Many people have trouble reasoning proportionally. Typically, when people first begin to think about proportions, they think in absolute, rather than in relative, terms. These different ways of thinking correspond to using additive (how much more is 12 than 10?) rather than multiplicative (what is the ratio of 10 to 20 compared to 12 to 25?) reasoning.

This session introduces the idea that there are different meanings of "more" and distinguishes between relative and absolute comparisons. To familiarize ourselves with the idea of equivalent ratios, we will use both additive and multiplicative methods to explore different ways of making similar figures. We will look at mixture problems and explore ratios without using algorithms to convert them to common denominators. Finally, we will examine characteristics of equations and graphs that represent direct variation.

ReviewGroups: Discuss any questions about the homework. If time allows, take a few minutes to try out the number games with a partner. Pairs should show their networks to one another. One partner can choose a (secret) input, run it through the network, and reveal only the output. Then the other partner can use the "undoing" network to find the original number.

Groups: Take a minute and discuss iteration, along with Problems H4 and H5 from Session 3. It is likely to be a new idea.